hölder singular metrics on big line bundles and equidistribution

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HOLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION

DAN COMAN, GEORGE MARINESCU, AND VIET-ANH NGUYEN

ABSTRACT. We show that normalized currents of integration along the common zeros ofrandom m-tuples of sections of powers of m singular Hermitian big line bundles on a

compact Kahler manifold distribute asymptotically to the wedge product of the curvature

currents of the metrics. If the Hermitian metrics are Holder with singularities we alsoestimate the speed of convergence.

CONTENTS

1. Introduction 12. Asymptotic behavior of Bergman kernel functions 53. Intersection of Fubini-Study currents and Bertini type theorem 64. Convergence speed towards intersection of Fubini-Study currents 104.1. Dinh-Sibony equidistribution theorem 104.2. Equidistribution of pullbacks of Dirac masses by Kodaira maps 115. Equidistribution with convergence speed for Holder singular metrics 18References 22

1. INTRODUCTION

Random polynomials or more generally holomorphic sections and the distribution oftheir zeros represent a classical subject in analysis [BP, ET, H, K], and they have beenmore recently used to model quantum chaotic eigenfunctions [BBL, NV].

This area witnessed intense activity recently [BL, BS, BMO, DMS, DS, S, SZ, ST], andespecially results about equidistribution of holomorphic sections in singular Hermitianholomorphic bundles were obtained [CM1, CM2, CM3, CMM, DMM] with emphasis onthe speed of convergence. The equidistribution is linked to the Quantum Unique Ergod-icity conjecture of Rudnick-Sarnak [RS], cf. [HS, Mar].

Date: May 25, 2015.2010 Mathematics Subject Classification. Primary 32L10; Secondary 32A60, 32U40, 32W20, 53C55,

81Q50.Key words and phrases. Bergman kernel function, Fubini-Study current, big line bundle, singular Her-

mitian metric, zeros of random holomorphic sections.D. Coman is partially supported by the NSF Grant DMS-1300157.

G. Marinescu is partially supported by DFG funded projects SFB/TR 12, MA 2469/2-2.V.-A. Nguyen is partially supported by Humboldt Foundation and Max-Planck Institute for Mathematics.

Funded through the Institutional Strategy of the University of Cologne within the German Excellence

Initiative.

1

http://de.arxiv.org/abs/1506.01727v1

HOLDER SINGULAR METRICS ON BIG LINE BUNDLES AND EQUIDISTRIBUTION 2

The equidistribution of common zeros of several sections is particularly interesting.Their study is difficult in the singular context and equidistribution with the estimate ofconvergence speed was established in [DMM] for Holder continuous metrics.

In this paper we obtain the equidistribution of common zeros of sections of m singularHermitian line bundles under the hypothesis that the metrics are continuous outsideanalytic sets intersecting generically. We will moreover introduce the notion of Holdermetric with singularities and establish the equidistribution with convergence speed ofcommon zeros.

Let (X,ω) be a compact Kahler manifold of dimension n and dist be the distance on Xinduced by ω. If (L, h) is a singular Hermitian holomorphic line bundle on X we denoteby c1(L, h) its curvature current. Recall that if eL is a holomorphic frame of L on someopen set U ⊂ X then |eL|2h = e−2φ, where φ ∈ L1

loc(U) is called the local weight of the

metric h with respect to eL, and c1(L, h)|U = ddcφ. Here d = ∂ + ∂, dc = 12πi

(∂ − ∂). Wesay that h is positively curved if c1(L, h) ≥ 0 in the sense of currents. This is equivalent tosaying that the local weights φ are plurisubharmonic (psh).

Recall that a holomorphic line bundle L is called big if its Kodaira-Iitaka dimensionequals the dimension of X (see [MM1, Definition 2.2.5]). By the Shiffman-Ji-Bonavero-Takayama criterion [MM1, Lemma 2.3.6], L is big if and only if it admits a singular metrich with c1(L, h) ≥ εω for some ε > 0.

Let (Lk, hk), 1 ≤ k ≤ m ≤ n, be m singular Hermitian holomorphic line bundles on

(X,ω). Let H0(2)(X,L

pk) be the Bergman space of L2-holomorphic sections of Lp

k := L⊗pk

relative to the metric hk,p := h⊗pk induced by hk and the volume form ωn on X, endowed

with the inner product

(1) (S, S ′)k,p :=

∫

X

〈S, S ′〉hk,pωn , S, S ′ ∈ H0

(2)(X,Lpk).

Set ‖S‖2k,p = (S, S)k,p, dk,p = dimH0(2)(X,L

pk) − 1. For every p ≥ 1 we consider the multi-

projective space

(2) Xp := PH0(2)(X,L

p1)× . . .× PH0

(2)(X,Lpm)

equipped with the probability measure σp which is the product of the Fubini-Study vol-umes on the components. If S ∈ H0(X,Lp

k) we denote by [S = 0] the current of integra-tion (with multiplicities) over the analytic hypersurface S = 0 of X. Set

[sp = 0] := [sp1 = 0] ∧ . . . ∧ [spm = 0] , for sp = (sp1, . . . , spm) ∈ Xp,

whenever this is well-defined (cf. Section 3). We also consider the probability space

(Ω, σ∞) :=∞∏

p=1

(Xp, σp) .

Let us recall the following:

Definition 1.1. We say that the analytic subsets A1, . . . , Am, m ≤ n, of a compact complexmanifold X of dimension n are in general position if codimAi1 ∩ . . . ∩ Aik ≥ k for every1 ≤ k ≤ m and 1 ≤ i1 < . . . < ik ≤ m.


Here is our first main result.

Theorem 1.2. Let (X,ω) be a compact Kahler manifold of dimension n and (Lk, hk), 1 ≤k ≤ m ≤ n, be m singular Hermitian holomorphic line bundles on X such that hk iscontinuous outside a proper analytic subset Σ(hk) ⊂ X, c1(Lk, hk) ≥ εω on X for someε > 0, and Σ(h1), . . . ,Σ(hm) are in general position. Then for σ∞-a.e. spp≥1 ∈ Ω, we havein the weak sense of currents on X,

1

pm[sp = 0] → c1(L1, h1) ∧ . . . ∧ c1(Lm, hm) as p→ ∞ .

In order to prove this theorem we show in Theorem 4.2 that the currents 1pm

[sp = 0]

distribute as p → ∞ like the wedge product of the normalized Fubini-Study currents ofthe spaces H0

(2)(X,Lpk) defined in (10) below. Then in Proposition 3.1 we prove that the

latter sequence of currents converges to c1(L1, h1) ∧ . . . ∧ c1(Lm, hm).Our second main result gives an estimate of the speed of convergence in Theorem 1.2

in the case when the metrics are Holder with singularities.

Definition 1.3. We say that a function φ : U → [−∞,∞) defined on an open subsetU ⊂ X is Holder with singularities along a proper analytic subset Σ ⊂ X if there existconstants c, > 0 and 0 < ν ≤ 1 such that

(3) |φ(z)− φ(w)| ≤ c dist(z, w)ν

mindist(z,Σ), dist(w,Σ)holds for all z, w ∈ U \ Σ. A singular metric h on L is called Holder with singularities alonga proper analytic subset Σ ⊂ X if all its local weights are Holder functions with singularitiesalong Σ.

Holder singular Hermitian metrics appear frequently in complex geometry and pluri-potential theory. Let us first observe that metrics with analytic singularities [MM1, Def-inition 2.3.9], which are very important for the regularization of currents and for tran-scendental methods in algebraic geometry [BD, CM3, D3, D4, D6], are Holder metricswith singularities. The class of Holder metrics with singularities is invariant under pull-back and push-forward by meromorphic maps. In particular, this class is invariant underbirational maps, e. g. blow-up and blow-down. They occur also as certain quasiplurisub-harmonic upper envelopes (e. g. Hermitian metrics with minimal singularities on a bigline bundle, equilibrium metrics, see [BB, DMM, DMN], especially [BD, Theorem 1.4]).

Theorem 1.4. In the setting of Theorem 1.2 assume in addition that hk is Holder withsingularities along Σ(hk) and set Σ := Σ(h1) ∪ . . . ∪ Σ(hm). Then there exist a constantξ > 0 depending only on m and a constant c = c(X,L1, h1, . . . , Lm, hm) > 0 with thefollowing property: For any sequence of positive numbers λpp≥1 such that

lim infp→∞

λplog p

> (1 + ξn)c,

there are subsets Ep ⊂ Xp such that for p large enough,

(a) σp(Ep) ≤ cpξn exp(−λp/c) ,


(b) if sp ∈ Xp \ Ep we have

∣∣∣⟨ 1

pm[sp = 0]−

m∧

k=1

c1(Lk, hk), φ⟩∣∣∣ ≤ c

(λpp

+1

p

∣∣∣ log dist(supp ddcφ,Σ)∣∣∣)‖φ‖C 2 ,

for any form φ of class C 2 such that ddcφ = 0 in a neighborhood of Σ .

Here suppψ denotes the support of the form ψ. Let Pp be the Bergman kernel functionof the space H0

(2)(X,Lp) defined in (4). The proof of Theorem 1.4 uses the estimate for Pp

obtained in Theorem 2.1 in the case when the metric h on L is Holder with singularities.One can also apply Theorem 2.1 to study the asymptotics with speed of common zeros

of random n-tuples of sections of a (single) big line bundle endowed with a Holder Her-mitian metric with isolated singularities. Let (L, h) be a singular Hermitian holomorphicline bundle on (X,ω) and H0

(2)(X,Lp) be the corresponding spaces of L2-holomorphic

sections. Consider the multi-projective space

X′p := (PH0

(2)(X,Lp))n

endowed with the product probability measure σ′p induced by the Fubini-Study volume

on PH0(2)(X,L

p), and let

(Ω′, σ′∞) :=

∞∏

p=1

(X′p, σ

′p) .

If sp = (sp1, . . . , spn) ∈ X′p we let [sp = 0] = [sp1 = 0] ∧ . . . ∧ [spn = 0], provided this

measure is well-defined.

Theorem 1.5. Let (X,ω) be a compact Kahler manifold of dimension n and (L, h) be asingular Hermitian holomorphic line bundle on X such that h is Holder with singularitiesin a finite set Σ = x1, . . . , xJ ⊂ X, and c1(L, h) ≥ εω for some ε > 0. Then there existC > 0, p0 ∈ N depending only on (X,ω, L, h), and subsets Ep ⊂ X

′p such that:

(a) σ′p(Ep) ≤ Cp−2 ;

(b) For every p > p0, every sp ∈ X′p \ Ep, and every function χ of class C 2 on X,

∣∣∣∣∫

X

χ

(1

pn[sp = 0]− c1(L, h)

n

)∣∣∣∣ ≤ Clog p

p1/3‖χ‖C 2 .

In particular, this estimate holds for σ′∞-a.e. sequence spp≥1 ∈ Ω′ provided that p is large

enough.

This paper is organized as follows. In Section 2 we prove a pointwise estimate forthe Bergman kernel function in the case of Holder metrics with singularities. Section3 is devoted to the study of the intersection of Fubini-Study currents and to a versionof the Bertini theorem. In Section 4 we consider the Kodaira map as a meromorphictransform and estimate the speed of convergence of the intersection of zero-divisors of mbundles. We use this to prove Theorem 1.2. Finally, in Section 5 we prove Theorem 1.4and Theorem 1.5.

Acknowledgment. We thank Vincent Guedj for useful discussions regarding this paper.


2. ASYMPTOTIC BEHAVIOR OF BERGMAN KERNEL FUNCTIONS

In this section we prove a theorem about the asymptotic behavior of the Bergmankernel function in the case when the metric is Holder with singularities.

Let (L, h) be a holomorphic line bundle over a compact Kahler manifold (X,ω) ofdimension n, where h is a singular Hermitian metric on L. Consider the space H0

(2)(X,Lp)

of L2-holomorphic sections of Lp relative to the metric hp := h⊗p induced by h andthe volume form ωn on X, endowed with the natural inner product (see (1)). Since

H0(2)(X,L

p) is finite dimensional, let Spj

dpj=0 be an orthonormal basis and denote by Pp

the Bergman kernel function defined by

(4) Pp(x) =

dp∑

j=0

|Spj (x)|2hp, |Sp

j (x)|2hp := 〈Spj (x), S

pj (x)〉hp, x ∈ X.

Note that this definition is independent of the choice of basis.

Theorem 2.1. Let (X,ω) be a compact Kahler manifold of dimension n and (L, h) be asingular Hermitian holomorphic line bundle on X such that c1(L, h) ≥ εω for some ε > 0.Assume that h is Holder with singularities along a proper analytic subset Σ of X and withparameters ν, as in (3). If Pp is the Bergman kernel function defined by (4) for the spaceH0

(2)(X,Lp), then there exist a constant c > 1 and p0 ∈ N which depend only on (X,ω, L, h)

such that for all z ∈ X \ Σ and all p ≥ p0

(5)1

c≤ Pp(z) ≤

cp2n/ν

dist(z,Σ)2n/ν·

Recall that by Theorem 5.3 in [CM1] we have limp→∞1plogPp(z) = 0 locally uniformly

on X \ Σ for any metric h which is only continuous outside of Σ. Theorem 2.1 refines[CM1, Theorem 5.3] in this context, and it is interesting to compare it to the asymptoticexpansion of the Bergman kernel function in the case of smooth metrics [B, C, HM, MM1,MM2, T, Z].

Proof. The proof follows from [CM1, Section 5], which is based on techniques of Demailly[D1, Proposition 3.1], [D6, Section 9]. Let x ∈ X and Uα ⊂ X be a coordinate neighbor-hood of x on which there exists a holomorphic frame eα of L. Let ψα be a psh weight ofh on Uα. Fix r0 > 0 so that the (closed) ball V := B(x, 2r0) ⋐ Uα and let U := B(x, r0).By [CM1, (7)] there exist constants c1 > 0, p0 ∈ N so that

− log c1p

≤ 1

plogPp(z) ≤

log(c1r−2n)

p+ 2

(maxB(z,r)

ψα − ψα(z)

)

holds for all p > p0, 0 < r < r0 and z ∈ U with ψα(z) > −∞.For z ∈ U \ Σ and r < mindist(z,Σ), r0 we have since ψα is Holder that

maxB(z,r)

ψα − ψα(z) ≤crν

(dist(z,Σ)− r),


where c > 0 depends only on x. Taking r = dist(z,Σ)/νp−1/ν < dist(z,Σ)/2 we obtain

− log c1 ≤ logPp(z) ≤ log c1 − 2n log r + 2+1cprν dist(z,Σ)−

= c0 + 2n log(dist(z,Σ)−/νp1/ν

).

This holds for all z ∈ U \ Σ and p > p0, with constants r0, p0, c0, c1 depending only on x.A standard compactness argument now finishes the proof.

3. INTERSECTION OF FUBINI-STUDY CURRENTS AND BERTINI TYPE THEOREM

In this section we show that the intersection of the Fubini-Study currents associatedto line bundles as in Theorem 1.2 is well-defined. Moreover, we show that the sequenceof wedge products of normalized Fubini-Study currents converges weakly to the wedgeproduct of the curvature currents of (Lk, hk). We then prove that almost all zero-divisorsof sections of large powers of these bundles are in general position in the sense of Defi-nition 1.1.

Let V be a vector space of complex dimension d+ 1. If V is endowed with a Hermitianmetric, then we denote by ω

FSthe induced Fubini-Study form on the projective space

P(V ) (see [MM1, pp. 65, 212]) normalized so that ωdFS

is a probability measure. We alsouse the same notations for P(V ∗).

We keep the hypotheses and notation of Theorem 1.2. Namely, (Lk, hk), 1 ≤ k ≤m ≤ n, are singular Hermitian holomorphic line bundles on the compact Kahler manifold(X,ω) of dimension n, such that hk is continuous outside a proper analytic subset Σ(hk) ⊂X, c1(Lk, hk) ≥ εω for some ε > 0, and Σ(h1), . . . ,Σ(hm) are in general position in thesense of Definition 1.1.

Consider the space H0(2)(X,L

pk) of L2-holomorphic sections of Lp

k endowed with the

inner product (1). Since c1(Lk, hk) ≥ εω, it is well-known that H0(2)(X,L

pk) is nontrivial

for p sufficiently large, see e. g. Proposition 4.7. Let

dk,p := dimH0(2)(X,L

pk)− 1.

The Kodaira map associated to (Lpk, hk,p) is defined by

(6) Φk,p : X 99K G(dk,p, H0(2)(X,L

pk)) , Φk,p(x) :=

s ∈ H0

(2)(X,Lpk) : s(x) = 0

,

where G(dk,p, H0(2)(X,L

pk)) denotes the Grassmannian of hyperplanes in H0

(2)(X,Lpk) (see

[MM1, p. 82]). Let us identify G(dk,p, H0(2)(X,L

pk)) with P(H0

(2)(X,Lpk)

∗) by sending a hy-

perplane to an equivalence class of non-zero complex linear functionals on H0(2)(X,L

pk)

having the hyperplane as their common kernel. By composing Φk,p with this identifica-tion, we obtain a meromorphic map

(7) Φk,p : X 99K P(H0(2)(X,L

pk)

∗).

To get an analytic description of Φk,p, let

(8) Sk,pj ∈ H0

(2)(X,Lpk), j = 0, . . . , dk,p ,

be an orthonormal basis and denote by Pk,p the Bergman kernel function of the spaceH0

(2)(X,Lpk) defined as in (4). This basis gives identifications H0

(2)(X,Lpk) ≃ Cdk,p+1 and


P(H0(2)(X,L

pk)

∗) ≃ Pdk,p. Let U be a contractible Stein open set in X, let ek be a local

holomorphic frame for Lk on U , and write Sk,pj = sk,pj e⊗p

k , where sk,pj is a holomorphic

function on U. By composing Φk,p given in (7) with the last identification, we obtain ameromorphic map Φk,p : X 99K Pdk,p which has the following local expression

(9) Φk,p(x) = [sk,p0 (x) : . . . : sk,pdk,p(x)] for x ∈ U.

It is called the Kodaira map defined by the basis Sk,pj dk,pj=0.

Next, we define the Fubini-Study currents γk,p of H0(2)(X,L

pk) by

(10) γk,p|U =1

2ddc log

dk,p∑

j=0

|sk,pj |2,

where the open set U and the holomorphic functions sk,pj are as above. Note that γk,p is a

positive closed current of bidegree (1, 1) on X, and is independent of the choice of basis.Actually, the Fubini-Study currents are pullbacks of the Fubini-Study forms by Kodairamaps, which justifies their name.

Let ωFS

be the Fubini-Study form on Pdk,p. By (9) and (10), the currents γk,p can bedescribed as pullbacks

(11) γk,p = Φ∗k,p(ωFS

), 1 ≤ k ≤ m.

We introduce the psh function

(12) uk,p :=1

2plog

dk,p∑

j=0

|sk,pj |2 = uk +1

2plogPk,p on U ,

where uk is the weight of the metric hk on U corresponding to ek, so |ek|hk= e−uk . Clearly,

by (10) and (12), ddcuk,p =1pγk,p. Moreover, note that by (12), logPk,p ∈ L1(X,ωn) and

(13)1

pγk,p = c1(Lk, hk) +

1

2pddc logPk,p

as currents on X. By [CM1, Theorem 5.1, Theorem 5.3] (see also [CM1, (7)]) there existc > 0, p0 ∈ N, such that if p ≥ p0, 1 ≤ k ≤ m and z ∈ X \Σ(hk), then Pk,p(z) ≥ c. By (12)it follows that

(14) uk,p(z) ≥ uk(z) +log c

2p, z ∈ U, p ≥ p0, 1 ≤ k ≤ m.

For p ≥ 1 consider the following analytic subsets of X:

Σk,p :=x ∈ X : Sk,p

j (x) = 0, 0 ≤ j ≤ dk,p

, 1 ≤ k ≤ m.

Hence Σk,p is the base locus of H0(2)(X,L

pk), and Σk,p ∩ U = uk,p = −∞. Note also that

Σ(hk) ∩ U ⊃ uk = −∞ and by (14) we have Σk,p ⊂ Σ(hk) for p ≥ p0.


Proposition 3.1. In the hypotheses of Theorem 1.2 we have the following:

(i) For all p sufficiently large and every J ⊂ 1, . . . , m the analytic sets Σk,p , k ∈ J , Σ(hℓ) ,ℓ ∈ J ′ := 1, . . . , m \ J , are in general position.

(ii) If p is sufficiently large then the currents∧

k∈J

γk,p ∧∧

ℓ∈J ′

c1(Lℓ, hℓ)

are well defined on X, for every J ⊂ 1, . . . , m.(iii) 1

pmγ1,p ∧ . . . ∧ γm,p → c1(L1, h1) ∧ . . . ∧ c1(Lm, hm) as p → ∞, in the weak sense of

currents on X.

Proof. As noted above we have by (14) that Σk,p ⊂ Σ(hk) for all p sufficiently large.Since Σ(h1), . . . ,Σ(hm) are in general position this implies (i). Then (ii) follows by [D5,Corollary 2.11].

(iii) Let U ⊂ X be a contractible Stein open set as above, uk,p, uk be the psh functionsdefined in (12), so ddcuk = c1(Lk, hk) and ddcuk,p = 1

pγk,p on U . By [CM1, Theorem

5.1] we have that 1plogPk,p → 0 in L1(X,ωn), hence by (12), uk,p → uk in L1

loc(U), as

p → ∞, for each 1 ≤ k ≤ m. Recall that by (14), uk,p ≥ uk − Cp

holds on U for all

p sufficiently large and some constant C > 0. Then [FS, Theorem 3.5] implies thatddcu1,p ∧ . . . ∧ ddcum,p → ddcu1 ∧ . . . ∧ ddcum weakly on U as p→ ∞.

We will need the following version of Bertini’s theorem. The corresponding statementfor the case of a single line bundle is proved in [CM1, Proposition 4.1].

Proposition 3.2. Let Lk −→ X, 1 ≤ k ≤ m ≤ n, be holomorphic line bundles over acompact complex manifold X of dimension n. Assume that:

(i) Vk is a vector subspace of H0(X,Lk) with basis Sk,0, . . . , Sk,dk, base locus BsVk :=Sk,0 = . . . = Sk,dk = 0 ⊂ X, such that dk ≥ 1 and the analytic sets Bs V1, . . . ,BsVm are ingeneral position in the sense of Definition 1.1.

(ii) Z(tk) := x ∈ X :∑dk

j=0 tk,jSk,j(x) = 0, where tk = [tk,0 : . . . : tk,dk ] ∈ Pdk .

(iii) ν = µ1 × . . . × µm is the product measure on Pd1 × . . . × Pdm , where µk is theFubini-Study volume on Pdk .

Then the analytic sets Z(t1), . . . , Z(tm) are in general position for ν-a.e. (t1, . . . , tm) ∈Pd1 × . . .× Pdm .

Proof. If 1 ≤ l1 < . . . < lk ≤ m let νl1...lk = µl1 × . . . × µlk be the product measure on

Pdl1 × . . .× P

dlk . For 1 ≤ k ≤ m consider the sets

Uk = (tl1 , . . . , tlk) ∈ Pdl1 × . . .× P

dlk : dimZ(tl1) ∩ . . . ∩ Z(tlk) ∩Aj ≤ n− k − j ,where 1 ≤ l1 < . . . < lk ≤ m, j = 0 and A0 = ∅, or 1 ≤ j ≤ m − k and Aj =BsVi1 ∩ . . . ∩ BsVij for some i1 < . . . < ij in 1, . . . , m \ l1, . . . , lk.

The proposition follows if we prove by induction on k that

νl1...lk(Uk) = 1


for every set Uk with 1 ≤ l1 < . . . < lk ≤ m, 0 ≤ j ≤ m − k and Aj as above. Clearly,it suffices to consider the case l1, . . . , lk = 1, . . . , k. To simplify notation we set νk :=ν1...k.

Let k = 1. If j = 0, A0 = ∅, so U1 = t1 ∈ Pd1 : dimZ(t1) ≤ n− 1 = Pd1 . Assume next

that 1 ≤ j ≤ m− 1 and write Aj =⋃N

l=1Dl ∪B, where Dl are the irreducible components

of Aj of dimension n − j and dimB ≤ n − j − 1. We have that t1 ∈ Pd1 : Dl ⊂ Z(t1)is a proper linear subspace of Pd1 . Indeed, otherwise Dl ⊂ BsV1, so dimAj ∩ Bs V1 =n − j, which contradicts the hypothesis that BsV1, . . . ,BsVm are in general position. Ift1 ∈ P

d1 \ U1 then dimZ(t1) ∩ Aj ≥ n − j. Since Z(t1) ∩ Aj is an analytic subset of Aj , it

follows that Dl ⊂ Z(t1) ∩ Aj for some l, hence Pd1 \ U1 =

⋃Nl=1t1 ∈ P

d1 : Dl ⊂ Z(t1).Therefore µ1(P

d1 \ U1) = 0.We assume now that νk(Uk) = 1 for any set Uk as above. Let

Uk+1 = (t1, . . . , tk+1) ∈ Pd1 × . . .×P

dk+1 : dimZ(t1)∩ . . .∩Z(tk+1)∩Aj ≤ n− k− 1− j ,where 0 ≤ j ≤ m−k−1, A0 = ∅, or Aj = BsVi1∩. . .∩BsVij with k+2 ≤ i1 < . . . < ij ≤ m.Consider the set U = U ′ ∩ U ′′, where

U ′ = (t1, . . . , tk) ∈ Pd1 × . . .× P

dk : dimZ(t1) ∩ . . . ∩ Z(tk) ∩ Aj ≤ n− k − j,

U ′′ = (t1, . . . , tk) ∈ Pd1 × . . .×P

dk : dimZ(t1)∩ . . .∩Z(tk)∩Bs Vk+1∩Aj ≤ n−k− j−1.By the induction hypothesis we have νk(U

′) = νk(U′′) = 1, so νk(U) = 1. To prove that

νk+1(Uk+1) = 1 it suffices to show that

νk+1(W ) = 0 , where W := (U × Pdk+1) \ Uk+1.

To this end we fix t := (t1, . . . , tk) ∈ U , we let

Z(t) := Z(t1)∩ . . .∩Z(tk) , W (t) := tk+1 ∈ Pdk+1 : dimZ(t)∩Aj ∩Z(tk+1) ≥ n− k − j,

and prove that µk+1(W (t)) = 0.

Since t ∈ U ⊂ U ′ we can write Z(t) ∩ Aj =⋃N

l=1Dl ∪ B, where Dl are the irreduciblecomponents of Z(t)∩Aj of dimension n−k− j and dimB ≤ n−k− j−1. If tk+1 ∈ W (t)then Z(t) ∩ Aj ∩ Z(tk+1) is an analytic subset of Z(t) ∩ Aj of dimension n − k − j, soDl ⊂ Z(t) ∩ Aj ∩ Z(tk+1) for some l. Thus

W (t) =

N⋃

l=1

Fl(t) , where Fl(t) := tk+1 ∈ Pdk+1 : Dl ⊂ Z(tk+1).

If Dl ⊂ BsVk+1 then dimZ(t) ∩ Aj ∩ BsVk+1 = n − k − j, which contradicts the factthat t ∈ U ′′. Hence the sections in Vk+1 cannot all vanish on Dl, so we may assume thatSk+1,dk+1

6≡ 0 on Dl. We have Fl(t) ⊂ tk+1,0 = 0 ∪Hl(t) where

Hl(t) := [1 : tk+1,1 : . . . : tk+1,dk+1] ∈ P

dk+1 : Dl ⊂ Z([1 : tk+1,1 : . . . : tk+1,dk+1]) .

For each (tk+1,1 : . . . : tk+1,dk+1−1) ∈ Cdk+1−1 there exists at most one ζ ∈ C with [1 : tk+1,1 :. . . : tk+1,dk+1−1 : ζ ] ∈ Hl(t). Indeed, if ζ 6= ζ ′ have this property then

Sk+1,0 + tk+1,1Sk+1,1 + . . .+ tk+1,dk+1−1Sk+1,dk+1−1 + aSk+1,dk+1≡ 0 on Dl ,


for a = ζ, ζ ′, hence Sk+1,dk+1≡ 0 on Dl, a contradiction. It follows that µk+1(Hl(t)) = 0,

so µk+1(Fl(t)) = 0. Hence µk+1(W (t)) = 0 and the proof is complete.

We return now to the setting of Theorem 1.2. If Sk,pj dk,pj=0 is an orthonormal basis of

H0(2)(X,L

pk), we define the analytic hypersurface Z(tk) ⊂ X, for tk = [tk,0 : . . . : tk,dk,p] ∈

Pdk,p , as in Proposition 3.2 (ii). Let µk,p be the Fubini-Study volume on Pdk,p , 1 ≤ k ≤ m,p ≥ 1, and let µp = µ1,p× . . .×µm,p be the product measure on Pd1,p × . . .×Pdm,p . ApplyingProposition 3.2 we obtain:

Proposition 3.3. In the above setting, if p is sufficiently large then for µp-a.e. (t1, . . . , tm) ∈Pd1,p × . . . × Pdm,p the analytic subsets Z(t1), . . . , Z(tm) ⊂ X are in general position, andZ(ti1) ∩ . . . ∩ Z(tik) has pure dimension n− k for each 1 ≤ k ≤ m, 1 ≤ i1 < . . . < ik ≤ m.

Proof. Let Vk,p := H0(2)(X,L

pk), so BsVk,p = Σk,p. By Proposition 3.1, Σ1,p, . . . ,Σm,p are

in general position for all p sufficiently large. We fix such p and denote by [Z(tk)] thecurrent of integration along the analytic hypersurface Z(tk); it has the same cohomologyclass as pc1(Lk, hk). Proposition 3.2 shows that the analytic subsets Z(t1), . . . , Z(tm) arein general position for µp-a.e. (t1, . . . , tm) ∈ Pd1,p × . . . × Pdm,p . Hence if 1 ≤ k ≤ m,1 ≤ i1 < . . . < ik ≤ m, the current [Z(ti1)]∧ . . .∧ [Z(tik)] is well defined by [D5, Corollary2.11] and it is supported in Z(ti1) ∩ . . . ∩ Z(tik). Since c1(Lk, hk) ≥ εω, it follows that∫

X

[Z(ti1)]∧ . . .∧ [Z(tik)]∧ωn−k = pk∫

X

c1(Li1 , hi1)∧ . . .∧ c1(Lik , hik)∧ωn−k > pkεk∫

X

ωn.

So Z(ti1) ∩ . . . ∩ Z(tik) 6= ∅, hence it has pure dimension n− k.

4. CONVERGENCE SPEED TOWARDS INTERSECTION OF FUBINI-STUDY CURRENTS

In this section we rely on techniques introduced by Dinh-Sibony [DS], based on thenotion of meromorphic transform, in order to estimate the speed of equidistribution ofthe common zeros of m-tuples of sections of the considered big line bundles towards theintersection of the Fubini-Study currents. We then prove Theorem 1.2.

4.1. Dinh-Sibony equidistribution theorem. A meromorphic transform F : X 99K Ybetween two compact Kahler manifolds (X,ω) of dimension n and (Y, ωY ) of dimensionm is the data of an analytic subset Γ ⊂ X × Y (called the graph of F ) of pure dimensionm + k such that the projections π1 : X × Y → X and π2 : X × Y → Y restrictedto each irreducible component of Γ are surjective. We set formally F = π2 (π1|Γ)−1.For y ∈ Y generic (that is, outside a proper analytic subset), the dimension of the fiberF−1(y) := π1(π

−12 |Γ(y)) is equal to k. This is called the codimension of F . We consider two

of the intermediate degrees for F (see [DS, Section 3.1]):

d(F ) :=

∫

X

F ∗(ωmY ) ∧ ωk and δ(F ) :=

∫

X

F ∗(ωm−1Y ) ∧ ωk+1.

By [DS, Proposition 2.2], there exists r := r(Y, ωY ) such that for every positive closedcurrent T of bidegree (1, 1) on Y with ‖T‖ = 1 there is a smooth (1, 1)-form α whichdepends uniquely on the class T and a quasiplurisubharmonic (qpsh) function ϕ such


that −rωY ≤ α ≤ rωY and ddcϕ − T = α. If Y is the projective space Pℓ equipped withthe Fubiny-Study form ω

FS, then we have r(Pℓ, ω

FS) = 1. Consider the class

Q(Y, ωY ) := ϕ qpsh on Y, ddcϕ ≥ −r(Y, ωY )ωY .A positive measure µ on Y is called a BP measure if all qpsh functions on Y are integrablewith respect to µ. When dim Y = 1, it is well-known that µ is BP if and only if it admitslocally a bounded potential. The terminology BP comes from this fact (see [DS]).

If µ is a BP measure on Y and t ∈ R, we let

R(Y, ωY , µ) := sup

maxY

ϕ : ϕ ∈ Q(Y, ωY ),

∫

Y

ϕdµ = 0

,

∆(Y, ωY , µ, t) := sup

µ(ϕ < −t) : ϕ ∈ Q(Y, ωY ),

∫

Y

ϕdµ = 0

.

Let Φp be a sequence of meromorphic transforms from a compact Kahler manifold(X,ω) into compact Kahler manifolds (Xp, ωp) of the same codimension k, where Xp isdefined in (2). Let νp be a BP probability measure on Xp and ν∞ =

∏p≥1 νp be the product

measure on Ω :=∏

p≥1Xp. For every p > 0 and ε > 0 let

Ep(ε) :=⋃

‖φ‖C2≤1

xp ∈ Xp :

∣∣⟨Φ∗p(δxp

)− Φ∗p(νp), φ

⟩∣∣ ≥ d(Φp)ε,

where δxpis the Dirac mass at xp. Note that Φ∗

p(δxp) and Φ∗

p(νp) are positive closed currents

of bidimension (k, k) on X, and the former is well defined for the generic point xp ∈Xp (see [DS, Section 3.1]). Now we are in position to state the part which deals withthe quantified speed of convergence in the Dinh-Sibony equidistribution theorem [DS,Theorem 4.1].

Theorem 4.1 ([DS, Lemma 4.2 (d)]). In the above setting the following estimate holds:

νp(Ep(ε)) ≤ ∆(Xp, ωp, νp, ηε,p

),

where ηε,p := εδ(Φp)−1d(Φp)− 3R(Xp, ωp, νp).

4.2. Equidistribution of pullbacks of Dirac masses by Kodaira maps. Let (X,ω) bea compact Kahler manifold of dimension n and (Lk, hk), 1 ≤ k ≤ m ≤ n, be singularHermitian holomorphic line bundles on X such that hk is continuous outside a properanalytic subset Σ(hk) ⊂ X, c1(Lk, hk) ≥ εω on X for some ε > 0, and Σ(h1), . . . ,Σ(hm)are in general position. Recall from Section 1 that

Xp := PH0(2)(X,L

p1)× . . .× PH0

(2)(X,Lpm) , (Ω, σ∞) :=

∞∏

p=1

(Xp, σp),

where the probability measure σp is the product of the Fubini-Study volume on each

factor. From now on let p ∈ N be large enough. Fix an orthonormal basis Sk,pj dk,pj=0 as

in (8) and let Φk,p : X 99K Pdk,p be the Kodaira map defined by this basis (see (9)).By (11) we have that Φ∗

k,pωFS= γk,p, where γk,p is the Fubini-Study current of the space

H0(2)(X,L

pk) as defined in (10).


We consider now the Kodaira maps as meromorphic transforms from X to PH0(2)(X,L

pk)

which we denote still by Φk,p : X 99K PH0(2)(X,L

pk). Precisely, this is the meromorphic

transform with graph

Γk,p =(x, s) ∈ X × PH0

(2)(X,Lpk) : s(x) = 0

, 1 ≤ k ≤ m.

Indeed, since dimH0(2)(X,L

pk) ≥ 2 (see e.g. Proposition 4.7 below), there exists, for every

x ∈ X, a section s ∈ H0(2)(X,L

pk) with s(x) = 0, so the projection Γk,p −→ X is surjective.

Moreover, since Lpk is non-trivial, every global holomorphic section of Lp

k must vanish atsome x ∈ X, hence the projection Γk,p −→ PH0

(2)(X,Lpk) is surjective. Note that

Φk,p(x) =s ∈ PH0

(2)(X,Lpk) : s(x) = 0

, Φ−1

k,p(s) =x ∈ X : s(x) = 0

.

Let Φp be the product transform of Φ1,p, . . . ,Φm,p (see [DS, Section 3.3]). It is the mero-morphic transform with graph

(15) Γp =(x, sp1, . . . , spm) ∈ X × Xp : sp1(x) = . . . = spm(x) = 0

.

By above, the projection Π1 : Γp −→ X is surjective. The second projection Π2 : Γp −→ Xp

is proper, hence by Remmert’s theorem Π2(Γp) is an analytic subvariety of Xp. Proposition3.3 implies that Π2(Γp) has full measure in Xp, so Π2 is surjective and Φp is a meromorphictransform of codimension n−m, with fibers

Φ−1p (sp) = x ∈ X : sp1(x) = . . . = spm(x) = 0 , where sp = (sp1, . . . , spm) ∈ Xp .

Considering the product transform of any Φi1,p, . . . ,Φik ,p, 1 ≤ i1 < . . . < ik ≤ m, andarguing as above it follows that, for sp = (sp1, . . . , spm) ∈ Xp generic, the analytic setssp1 = 0, . . . , spm = 0 are in general position. Hence by [D5, Corollary 2.11] thefollowing current of bidegree (m,m) is well defined on X:

Φ∗p(δsp) = [sp = 0] = [sp1 = 0] ∧ . . . ∧ [spm = 0] = Φ∗

1,p(δsp1) ∧ . . . ∧ Φ∗m,p(δspm) .

The main result of this section is the following theorem.

Theorem 4.2. Under the hypotheses of Theorem 1.2 there exist a constant ξ > 0 dependingonly on m and a constant c = c(X,L1, h1, . . . , Lm, hm) > 0 with the following property: Forany sequence of positive numbers λpp≥1 with

lim infp→∞

λplog p

> (1 + ξn)c,

there are subsets Ep ⊂ Xp such that

(a) σp(Ep) ≤ cpξn exp(−λp/c) for all p large enough;

(b) if sp ∈ Xp \ Ep we have that the estimate∣∣∣∣1

pm⟨[sp = 0]− γ1,p ∧ . . . ∧ γm,p , φ

⟩∣∣∣∣ ≤ cλpp‖φ‖C 2

holds for every (n−m,n−m) form φ of class C 2.In particular, for σ∞-a.e. s ∈ Ω the estimate from (b) holds for all p sufficiently large.


Prior to the proof we need to establish some preparatory results. Let

d0,p = d1,p + . . .+ dm,p

be the dimension of Xp and πk be the canonical projection of Xp onto its k-th factor. Let

ωp := cp(π∗1ωFS

+ . . .+ π∗mωFS

), so σp = ωd0,p

p .

Here ωFS

denotes, as usual, the Fubini-Study form on each factor PH0(2)(X,L

pk), and the

constant cp is chosen so that σp is a probability measure on Xp, thus

(16) (cp)−d0,p =

d0,p!

d1,p! . . . dm,p!.

Lemma 4.3. There is a constant c0 > 0 such that cp ≥ c0 for all p ≥ 1.

Proof. Fix p ≥ 1 large enough. For each 1 ≤ k ≤ m, let lk := dk,p. Using Stirling’s

formula ℓ! ≈ (ℓ/e)ℓ√2πℓ it suffices to show that there is a constant c > 0 such that for all

l1, . . . , lm ≥ 1,

log (l1 + . . .+ lm)−( l1 log l1l1 + . . .+ lm

+ . . .+lm log lm

l1 + . . .+ lm

)≤ c.

Since the function t 7→ t log t, t > 0, is convex, we infer that

1

m

(l1 log l1 + . . .+ lm log lm

)≥ l1 + . . .+ lm

mlog

l1 + . . .+ lmm

.

This implies the required estimate with c := logm.

Following subsection 4.1 we consider two intermediate degrees for the Kodaira mapsΦp:

dp = d(Φp) :=

∫

X

Φ∗p(ω

d0,pp ) ∧ ωn−m and δp = δ(Φp) :=

∫

X

Φ∗p(ω

d0,p−1p ) ∧ ωn−m+1.

The next result gives the asymptotic behavior of dp and δp as p→ ∞.

Lemma 4.4. We have dp = pm‖c1(L1, h1) ∧ . . . ∧ c1(Lm, hm)‖ and

δp =pm−1

cp

m∑

k=1

dk,pd0,p

∥∥∥m∧

l=1,l 6=k

c1(Ll, hl)∥∥∥ ≤ Cpm−1 ,

where C > 0 is a constant depending on (Lk, hk) , 1 ≤ k ≤ m.

Proof. We use a cohomological argument. For the first identity we replace ωd0,pp by a Dirac

mass δs, where s := (s1, . . . , sm) ∈ Xp is such that s1 = 0, . . . , sm = 0 are in generalposition, so the current Φ∗

p(δs) = [s1 = 0] ∧ . . . ∧ [sm = 0] is well defined (see Proposition

3.3). By the Poincare -Lelong formula [MM1, Theorem 2.3.3],

[sk = 0] = pc1(Lk, hk) + ddc log |sk|hk,p, 1 ≤ k ≤ m.

Since the current c1(L1, h1) ∧ . . . ∧ c1(Lm, hm) is well defined (see Proposition 3.1) itfollows that∫

X

Φ∗p(δs)∧ωn−m = pm

∫

X

θ1∧ . . .∧θm∧ωn−m = pm∫

X

c1(L1, h1)∧ . . .∧c1(Lm, hm)∧ωn−m,


where θk is a smooth closed (1, 1) form in the cohomology class of c1(Lk, hk). Thus

dp =

∫

X

Φ∗p(ω

d0,pp ) ∧ ωn−m =

∫

X

Φ∗p(δs) ∧ ωn−m = pm‖c1(L1, h1) ∧ . . . ∧ c1(Lm, hm)‖.

For the second identity, a straightforward computation shows that

ωd0,p−1p =

m∑

k=1

cd0,p−1p (d0,p − 1)!

d1,p! . . . (dk,p − 1)! . . . dm,p!π∗1ω

d1,pFS

∧ . . . ∧ π∗kω

dk,p−1FS

∧ . . . ∧ π∗mω

dm,p

FS.

Using (16) and replacing ωdk,pFS

(resp. ωdk,p−1FS

) by a generic point (resp. a generic complex

line) in PH0(2)(X,L

pk), we may replace ω

d0,p−1p by a current of the form

T :=

m∑

k=1

dk,pcpd0,p

[s1 × . . .×Dk × . . .× sm].

Here, Dk is a generic complex line in PH0(2)(X,L

pk) and (s1, . . . , sm) is a generic point in

Xp. The genericity of Dk implies that Φ∗k,p(Dk) = X, so

Φ∗p([s1 × . . .×Dk × . . .× sm]) =

m∧

l=1,l 6=k

[sl = 0] .

The Poincare-Lelong formula yields

‖Φ∗p([s1 × . . .×Dk × . . .× sm])‖ = pm−1

∥∥∥m∧

l=1,l 6=k

c1(Ll, hl)∥∥∥.

Since δp = ‖Φ∗p(T )‖, the second identity follows. Using Lemma 4.3, this yields the upper

bound on δp.

Lemma 4.5. For all p sufficiently large we have Φ∗p(σp) = γ1,p ∧ . . . ∧ γm,p .

Proof. Let us write Xp = X1,p × . . . × Xm,p and σp = σ1,p × . . . × σm,p, where Xk,p =PH0

(2)(X,Lpk) and σk,p is the Fubini-Study volume on Xk,p. Recall that the meromorphic

transform Φp has graph Γp defined in (15), and Π1 : Γp −→ X, Π2 : Γp −→ Xp, denotethe canonical projections. By the definition of Φ∗

p(σp) (see [DS, Sect. 3.1]) we have

〈Φ∗p(σp), φ〉 =

∫

Γp

Π∗1(φ) ∧ Π∗

2(σp) =

∫

Xp

Π2∗Π∗1(φ) ∧ σp =

∫

Xp

〈[sp = 0], φ〉 dσp(sp),

where φ is a smooth (n−m,n −m) form on X. Thanks to Propositions 3.1 and 3.2, wecan apply [CM1, Proposition 4.2] as in the proof of [CM1, Theorem 1.2] to show that

〈Φ∗p(σp), φ〉 =

∫

Xm,p

. . .

∫

X1,p

〈[sp1] = 0] ∧ . . . ∧ [spm = 0], φ〉 dσ1,p(sp1) . . . dσm,p(spm)

=

∫

Xm,p

. . .

∫

X2,p

〈γ1,p ∧ [sp2] = 0] ∧ . . . ∧ [spm = 0], φ〉 dσ2,p(sp2) . . . dσm,p(spm)

= . . . = 〈γ1,p ∧ . . . ∧ γm,p, φ〉 .This concludes the proof of the lemma.


Lemma 4.6. There exist absolute constants C1, α > 0, and constants C2, α′, ξ > 0 depending

only on m ≥ 1, such that for all ℓ, ℓ1, . . . , ℓm ≥ 1 and t ≥ 0,

R(Pℓ, ωFS, ωℓ

FS) ≤ 1

2(1 + log ℓ) ,

∆(Pℓ, ωFS, ωℓ

FS, t) ≤ C1ℓ e

−αt ,

r(Pℓ1 × . . .× Pℓm , ω

MP) ≤ r(ℓ1, . . . , ℓm) := max

1≤k≤m

d

ℓk,

R(Pℓ1 × . . .× Pℓm, ω

MP, ωd

MP) ≤ C2r(ℓ1, . . . , ℓm)(1 + log d) ,

∆(Pℓ1 × . . .× Pℓm, ω

MP, ωd

MP, t) ≤ C2d

ξe−α′t/r(ℓ1,...,ℓm) ,

where

d = ℓ1 + . . .+ ℓm , ωMP:= c

(π∗1(ωFS

) + . . .+ π∗m(ωFS

)), c−d =

d!

ℓ1! . . . ℓm!,

so ωdMP

is a probability measure on Pℓ1 × . . .× Pℓm.

Proof. The first two inequalities are proved in Proposition A.3 and Corollary A.5 from[DS]. If T is a positive closed current of bidegree (1, 1) on Pℓ1 × . . .× Pℓm with ‖T‖ = 1then T is in the cohomology class of α = a1π

∗1(ωFS

) + . . . + amπ∗m(ωFS

), for some ak ≥ 0.Hence

0 ≤ α ≤(

max1≤k≤m

akc

)ω

MP.

Now

1 = ‖T‖ =

∫

Pℓ1×...×Pℓm

α ∧ ωd−1MP

=m∑

k=1

akℓkcd

,

so ak/c ≤ d/ℓk. Thus r(Pℓ1 × . . . × Pℓm, ωMP

) ≤ max1≤k≤mdℓk

. The last two inequalities

follow from these estimates by applying [DS, Proposition A.8, Proposition A.9].

We will also need the following lower estimate for the dimension dk,p.

Proposition 4.7. Let (X,ω) be a compact Kahler manifold of dimension n. Let (L, h) → Xbe a singular Hermitian holomorphic line bundle such that c1(L, h) ≥ εω for some ε > 0 andh is continuous outside a proper analytic subset of X. Then there exists C > 0 and p0 ∈ N

such that

dimH0(2)(X,L

p) ≥ Cpn , ∀ p ≥ p0.

Proof. Let Σ ⊂ X be a proper analytic set such that h is continuous on X \ Σ. We fixx0 ∈ X \ Σ and r > 0 such that B(x0, 2r) ∩ Σ = ∅. Let 0 ≤ χ ≤ 1 be a smooth cut-

off function which equals 1 on B(x0, r) and is supported in B(x0, 2r). We consider thefunction ψ : X → [−∞,∞), ψ(x) = ηχ(x) log |x− x0|, where η > 0.

Consider the metric h0 = h exp(−ψ) on L. We choose η sufficiently small such that

c1(L, h0) ≥ε

2ω on X.


Let us denote by I(hp) the multiplier ideal sheaf associated to hp. Note that H0(2)(X,L

p) =

H0(X,Lp ⊗ I(hp)). The Nadel vanishing theorem [D6, N] shows that there exists p0 ∈ N

such that

(17) H1(X,Lp ⊗ I(hp0)) = 0 , p ≥ p0.

Note that I(hp0) = I(hp)⊗ I(pψ). Consider the exact sequence

(18) 0 → Lp ⊗ I(hp)⊗ I(pψ) → Lp ⊗ I(hp) → Lp ⊗ I(hp)⊗OX/I(pψ) → 0 .

Thanks to (17) applied to the long exact cohomology sequence associated to (18) wehave

(19) H0(X,Lp ⊗ I(hp)) → H0(X,Lp ⊗ I(hp)⊗OX/I(pψ)) → 0 , p ≥ p0.

Now, for x 6= x0, I(pψ)x = OX,x hence OX,x/I(pψ)x = 0. Moreover I(hp)x0= OX,x0

sinceh is continuous at x0. Hence

H0(X,Lp ⊗ I(hp)⊗OX/I(pψ)) = Lpx0

⊗ I(hp)x0⊗OX,x0

/I(pψ)x0

= Lpx0

⊗OX,x0/I(pψ)x0

,(20)

so

(21) H0(X,Lp ⊗ I(hp)) → Lpx0

⊗OX,x0/I(pψ)x0

→ 0 , p ≥ p0.

Denote by MX,x0the maximal ideal of OX,x0

(that is, germs of holomorphic functions

vanishing at x0). We have I(pψ)x0⊂ M[pη]−n+1

X,x0and dimOX,x0

/Mk+1X,x0

=(k+nk

), which

together with (21) implies the conclusion.

Proof of Theorem 4.2. We will apply Theorem 4.1 to the meromorphic transforms Φp fromX to the multi-projective space (Xp, ωp) defined above, and the BP measures νp := σp onXp. For t ∈ R and ε > 0 let

Rp := R(Xp, ωp, σp) , ∆p(t) := ∆(Xp, ωp, σp, t) ,

Ep(ε) :=⋃

‖φ‖C2≤1

s ∈ Xp :

∣∣⟨[s = 0]− γ1,p ∧ . . . ∧ γm,p , φ⟩∣∣ ≥ dpε

.(22)

It follows from Siegel’s lemma [MM1, Lemma 2.2.6] and Proposition 4.7 that there existsC3 > 0 depending only on (X,Lk, hk)1≤k≤m and p0 ∈ N such that

pn/C3 ≤ dk,p ≤ C3pn , p ≥ p0 , 1 ≤ k ≤ m.

By the last two inequalities in Lemma 4.6 we obtain for p ≥ p0 and t ≥ 0,

Rp ≤ mC2C23(1 + log(mC3p

n)) ≤ C4 log p ,

∆p(t) ≤ C2(mC3pn)ξ exp

(−α′t

mC23

)≤ C4p

ξne−t/C4 ,(23)

where C4 is a constant depending only on (X,Lk, hk)1≤k≤m. Now set

εp := λp/p , ηp := εpdp/δp − 3Rp .

Lemma 4.4 implies that dp ≈ pm, δp . pm−1, so

ηp ≥ C5λp − 3C4 log p , p ≥ p0 ,


where C5 is a constant depending only on (X,Lk, hk)1≤k≤m. Note that for all p sufficientlylarge,

ηp >C5

2λp , provided that lim inf

p→∞

λplog p

> 6C4/C5 .

If Ep = Ep(εp) then it follows from Theorem 4.1 and Lemma 4.5 that for all p sufficientlylarge

σp(Ep) ≤ ∆p(ηp) ≤ C4pξn exp

(−C5λp2C4

),

where for the last estimate we used (23). Let

c = max

(6C4

C5(1 + ξn),2C4

C5, C4, ‖c1(L1, h1) ∧ . . . ∧ c1(Lm, hm)‖

).

If lim infp→∞(λp/ log p) > (1 + ξn)c then for all p sufficiently large

σp(Ep) ≤ C4pξn exp

(−C5λp2C4

)≤ c pξn exp

(−λpc

).

On the other hand we have by the definition of Ep that if sp ∈ Xp \ Ep and φ is a (n −m,n−m) form of class C

2 then∣∣∣∣1

pm〈[sp = 0]− γ1,p ∧ . . . ∧ γm,p , φ〉

∣∣∣∣ ≤dppm

λpp‖φ‖C 2 ≤ c

λpp‖φ‖C 2 .

In the last inequality we used the fact that dp ≤ c pm by Lemma 4.4.For the last conclusion of Theorem 4.2 we proceed as in [DMM, p. 9]. The assumption

on λp/ log p and (a) imply that

∞∑

p=1

σp(Ep) ≤ c′∞∑

p=1

1

pη<∞

for some c′ > 0 and η > 1. Hence the set

E :=s = (s1, s2, . . .) ∈ Ω : sp ∈ Ep for infinitely many p

satisfies σ∞(E) = 0. Indeed, for every N ≥ 1, E is contained in the sets = (s1, s2, . . .) ∈ Ω : sp ∈ Ep for at least one p ≥ N

,

whose σ∞-measure is at most

∞∑

p=N

σp(Ep) ≤ c′∞∑

p=N

1

pη→ 0 as N → ∞.

The proof of the theorem is thereby completed.

Proof of Theorem 1.2. Theorem 1.2 follows directly from Theorem 4.2 and Proposition3.1 (iii).


5. EQUIDISTRIBUTION WITH CONVERGENCE SPEED FOR HOLDER SINGULAR METRICS

In this section we prove Theorems 1.4 and 1.5. We close with more examples of Holdermetrics with singularities. Theorem 1.4 follows at once from Theorem 4.2 and the nextresult.

Theorem 5.1. In the setting of Theorem 1.4, there exists a constant c > 0 depending onlyon (X,L1, h1, . . . , Lm, hm) such that for all p sufficiently large the estimate

∣∣∣⟨ 1

pm

m∧

k=1

γk,p −m∧

k=1

c1(Lk, hk), φ⟩∣∣∣ ≤ c

( log pp

+1

p

∣∣∣ log dist(supp ddcφ,Σ)∣∣∣)‖φ‖C 2

holds for every (n−m,n−m) form φ of class C 2 which satisfies ddcφ = 0 in a neighborhoodof Σ.

Proof. If φ is as in the statement, then using Proposition 3.1 and (13) we can write

⟨ 1

pm

m∧

k=1

γk,p −m∧

k=1

c1(Lk, hk), φ⟩=

m∑

k=1

Ik ,

where

Ik =⟨c1(L1, h1) ∧ . . . ∧ c1(Lk−1, hk−1) ∧

(γk,pp

− c1(Lk, hk))∧ γk+1,p

p∧ . . . ∧ γm,p

p, φ

⟩

=⟨c1(L1, h1) ∧ . . . ∧ c1(Lk−1, hk−1) ∧

ddc logPk,p

2p∧ γk+1,p

p∧ . . . ∧ γm,p

p, φ

⟩.

Since logPk,p is continuous on supp ddcφ we have

Ik =⟨c1(L1, h1) ∧ . . . ∧ c1(Lk−1, hk−1) ∧

γk+1,p

p∧ . . . ∧ γm,p

p,logPk,p

2pddcφ

⟩.

By (13) the masses ofγℓ,pp

and c1(Lℓ, hℓ) are equal. Moreover, by Theorem 2.1 there exist

a constant c′ > 1 and p0 ∈ N such that for all z ∈ X \ Σ and all p ≥ p0 one has

−c′ ≤ logPk,p ≤ c′ log p+ c′∣∣ log dist(z,Σ)

∣∣ , 1 ≤ k ≤ m.

It follows that

|Ik| ≤c

m

( log pp

+1

p

∣∣∣ log dist(supp ddcφ,Σ)∣∣∣)‖φ‖C 2 , ∀ p ≥ p0 , 1 ≤ k ≤ m,

with some constant c = c(X,L1, h1, . . . , Lm, hm) > 0, and the proof is complete.

Finally, we turn our attention to the proof of Theorem 1.5. To this end we need thefollowing:

Theorem 5.2. Let (X,ω) be a compact Kahler manifold of dimension n and (L, h) be asingular Hermitian holomorphic line bundle on X such that h is Holder with singularitiesin a finite set Σ = x1, . . . , xJ ⊂ X, and c1(L, h) ≥ εω for some ε > 0. Let γp be the


Fubini-Study current of H0(2)(X,L

p) defined in (10). Then there exist a constant C > 0 and

p0 ∈ N which depend only on (X,ω, L, h) such that∣∣∣∣∫

X

χ

(1

pnγnp − c1(L, h)

n

)∣∣∣∣ ≤ Clog p

p1/3‖χ‖C 2 ,

for every p > p0 and every function χ of class C 2 on X.

Proof. Throughout the proof we will denote by C > 0 a constant that depends only on(X,ω, L, h) and may change from an estimate to the next. Let r ∈ (0, 1) be a number to

be chosen later such that there exist coordinate balls B(xj , 2r) centered at xj , 1 ≤ j ≤ J ,which are disjoint. Let 0 ≤ χj ≤ 1, 1 ≤ j ≤ J , be a smooth function with suppχj ⊂B(xj , 2r), χj = 1 on B(xj , r), and set χ0 = 1−∑J

j=1 χj . Then

(24) ‖χj‖C 2 ≤ C/r2 , 0 ≤ j ≤ J ,

for some constant C > 0.Let now χ be a function of class C 2 on X. Using its Taylor expansion near xj we have

|χ(x)− χ(xj)| ≤ Cr‖χ‖C 1 for x ∈ B(xj , 2r), hence

(25) |χ(x)χj(x)− χ(xj)χj(x)| ≤ Cr‖χ‖C 2 χj(x) , ∀ x ∈ X .

If Pp is the Bergman kernel function of the space H0(2)(X,L

p) defined in (4) then using

(13) we obtain

1

pnγnp − c1(L, h)

n =1

2pddc logPp ∧Rp , where Rp =

n−1∑

ℓ=0

(γpp

)n−1−ℓ

∧ c1(L, h)ℓ.

Using a similar argument to that in the proof of Proposition 3.1 one shows that all ofthese currents are well defined (see also [CM1, Lemma 3.3 and Remark 3.5]). Moreover,

‖Rp‖ =

∫

X

Rp ∧ ω = n

∫

X

c1(L, h)n−1 ∧ ω ,

∫

X

1

pnγnp =

∫

X

c1(L, h)n .

Set Xr = X \⋃Jj=1B(xj , r). By Theorem 2.1 there exist C > 0 and p0 depending only on

(X,ω, L, h) such that for p > p0,

(26) ‖ logPp‖L∞(Xr) ≤ C(log p− log r) .

Since Rp is closed we have

∫

X

χ

(1

pnγnp − c1(L, h)

n

)=

J∑

j=0

∫

X

χχj

(1

pnγnp − c1(L, h)

n

)=(27)

1

2p

∫

X

(logPp)Rp ∧ ddc(χχ0) +J∑

j=1

∫

X

χχj

(1

pnγnp − c1(L, h)

n

).

Note that χ0 = 0 on⋃J

j=1B(xj , r), so we deduce by (24) and (26) that

(28)1

2p

∣∣∣∣∫

X

(logPp)Rp ∧ ddc(χχ0)

∣∣∣∣ ≤C‖χ‖C 2

pr2(log p− log r) .


For j ≥ 1 we obtain using (25),∣∣∣∣∫

X

χχj

(1

pnγnp − c1(L, h)

n

)∣∣∣∣ ≤∣∣∣∣∫

X

χ(xj)χj

(1

pnγnp − c1(L, h)

n

)∣∣∣∣+

+

∫

X

|χχj − χ(xj)χj |(

1

pnγnp + c1(L, h)

n

)

≤ |χ(xj)|2p

∣∣∣∣∫

X

(logPp)Rp ∧ ddcχj

∣∣∣∣+ Cr‖χ‖C 2 .

Since ddcχj is supported on Xr we use again (24) and (26) to get

(29)

∣∣∣∣∫

X

χχj

(1

pnγnp − c1(L, h)

n

)∣∣∣∣ ≤C‖χ‖C 2

pr2(log p− log r) + Cr‖χ‖C 2 ,

for 1 ≤ j ≤ J . By (27), (28), (29) we conclude that∣∣∣∣∫

X

χ

(1

pnγnp − c1(L, h)

n

)∣∣∣∣ ≤ C‖χ‖C 2

(log p− log r

pr2+ r

).

The proof is finished by choosing r = p−1/3 in the last estimate.

Proof of Theorem 1.5. This follows directly from Theorem 5.2 and Theorem 4.2. Indeed,one applies Theorem 4.2 with m = n, (Lk, hk) = (L, h), and for the sequence λp =(2 + ξn)c log p.

Let us close the paper with more examples of Holder metrics with singularities.

(1) Consider a projective manifold X and a smooth divisor Σ ⊂ X . By [Ko, TY], ifL = KX⊗OX(Σ) is ample, there exist a complete Kahler-Einstein metric ω on M := X \Σwith Ricω = −ω. This metric has Poincare type singularities, describe as follows. Wedenote by D the unit disc in C. Each x ∈ Σ has a coordinate neighborhood Ux such that

Ux∼= D

n, x = 0, Ux ∩ Σ ∼= z = (z1, . . . , zn) : z1 = 0, Ux ∩M ∼= D⋆ × D

n−1.

Then ω = i2

∑nj,k=1 gjkdzj ∧ dzk is equivalent to the Poincare type metric

ωP =i

2

dz1 ∧ dz1|z1|2(log |z1|2)2

+i

2

n∑

j=2

dzj ∧ dzj .

Let σ be the canonical section of OX(Σ) (cf. [MM1, p. 71]) and denote by hσ the metricinduced by σ on OX(Σ) (cf. [MM1, Example 2.3.4]). Note also that c1(OX(Σ), hσ) = [Σ]by [MM1, (2.3.8)]. Consider the metric

(30) hM,σ := hKM ⊗ hσ on L |M= KM ⊗ OX(Σ) |M∼= KM .

Note that L is trivial over Ux and the metric hM,σ has a weight ϕ on Ux ∩M ∼= D⋆ ×Dn−1

given by e2ϕ = |z1|2 det[gjk]. So ddcϕ = − 12π

Ricω > 0 and ϕ is psh on Ux ∩ M . Wesee as in [CM1, Lemma 6.8] that ϕ extends to a psh function on Ux, and hM,σ extendsuniquely to a positively curved metric hL on L. By construction, hL is a Holder metricwith singularities.


(2) Let us specialize the previous example to the case of Riemann surfaces. Let X be acompact Riemann surface of genus g and let Σ = p1, . . . , pd ⊂ X. It is well-known thatthe following conditions are equivalent:

(i) U = X \ Σ admits a complete Kahler-Einstein metric ω with Ricω = −ω,(ii) 2g − 2 + d > 0,(iii) L = KX ⊗ OX(Σ) is ample,(iv) the universal cover of U is the upper-half plane H.

If one of these equivalent conditions is satisfied, the Kahler-Einstein metric ω is inducedby the Poincare metric on H. In local coordinates z centered at p ∈ Σ we have ω =i2gdz ∧ dz where g has singularities of type |z|−2(log |z|2)−2. Note that ω extends to a

closed strictly positive (1, 1)-current on X. By [CM1, Lemma 6.8] there exists a singularmetric hL on L such that c1(L, h

L) = 12πω on X. The weight of hL near a point p ∈ Σ has

the form ϕ = 12log(|z|2g), which is Holder with singularities.

(3) Let X be a complex manifold, (L, hL0 ) a holomorphic line bundle on X with smoothHermitian metric such that c1(L, h

L0 ) is a Kahler metric. Let Σ be a compact divisor with

normal crossings. Let Σ1, . . . ,ΣN be the irreducible components of Σ, so Σj is a smoothhypersurface in X. Let σj be holomorphic sections of the associated holomorphic linebundle OX(Σj) vanishing to first order on Σj and let | · |j be a smooth Hermitian metric onOX(Σj) so that |σj |j < 1 and |σj |j = 1/e outside a relatively compact open set containingΣ. Set

Θδ = Ω + δddcF, where δ > 0 , F = −1

2

N∑

j=1

log(− log |σj|j).

For δ sufficiently small Θδ defines the generalized Poincare metric [MM1, Lemma 6.2.1],[CM1, Section 2.3]. For ε > 0,

hLε = hL0

N∏

j=1

(− log |σj |j)ε

is a singular Hermitian metric on L which is Holder with singularities. The curvaturec1(L, h

Lε ) is a strictly positive current on X, provided that ε is sufficiently small (cf. [MM1,

Lemma 6.2.1]). When X is compact the curvature current of hε dominates a small mul-tiple of Θδ on X \ Σ.

(4) Let X be a Fano manifold. Fix a Hermitian metric h0 on K−1X such that ω :=

c1(K−1X , h0) is a Kahler metric. We denote by PSH(X,ω) the set of ω-plurisubharmonic

functions on X. Let Σ be a smooth divisor in the linear system defined by K−ℓX , so there

exists a section s ∈ H0(X,K−ℓX ) with Σ = Div(s).

Fix a smooth metric h on the bundle OX(Σ) and let β ∈ [0, 1). A conic Kahler metric ωon X with cone angle β along Σ, cf. [Do, T2], is a current ω = ωϕ = ω + ddcϕ ∈ c1(X)

where ϕ = ψ + |s|2βh ∈ PSH(X,ω) and ψ ∈ C ∞(X) ∩ PSH(X,ω). In a neigbourhoodof a point of Σ where Σ is given by z1 = 0 the metric ω is equivalent to the cone metrici2(|z1|2β−2dz1 ∧ dz1 +

∑nj=2 dzj ∧ dzj).


The metric ω defines a singular metric hω on K−1X which is Holder with singularities.

Its curvature current is Ric ω := c1(K−1X , hω) = (1 − ℓβ)ω + β[Σ], where [Σ] is the current

of integration on Σ.

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DEPARTMENT OF MATHEMATICS, SYRACUSE UNIVERSITY, SYRACUSE, NY 13244-1150, USA

E-mail address: [email protected]

UNIVERSITAT ZU KOLN, MATHEMATISCHES INSTITUT, WEYERTAL 86-90, 50931 KOLN, DEUTSCHLAND &

INSTITUTE OF MATHEMATICS ‘SIMION STOILOW ’, ROMANIAN ACADEMY, BUCHAREST, ROMANIA


MATHEMATIQUE-BATIMENT 425, UMR 8628, UNIVERSITE PARIS-SUD, 91405 ORSAY, FRANCE


hölder singular metrics on big line bundles and equidistribution

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